What is the general solution for 5cot^2(x)+8cot(x)+3=0 ?

The general solution for 5cot^2(x)+8cot(x)+3=0 is x=2.11121…+pin,x=(3pi)/4+pin

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5cot^2(x)+8cot(x)+3=0

Solution

5 cot^{2} (x) + 8 cot (x) + 3 = 0

Solution

x = 2.11121 \dots + πn, x = \frac{3 π}{4} + πn

Degrees

x = 120.96375 \dots^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

5 cot^{2} (x) + 8 cot (x) + 3 = 0

Solve by substitution

5 cot^{2} (x) + 8 cot (x) + 3 = 0

Let:

cot (x) = u

5 u^{2} + 8 u + 3 = 0

5 u^{2} + 8 u + 3 = 0 : u = - \frac{3}{5}, u = - 1

5 u^{2} + 8 u + 3 = 0

Solve with the quadratic formula

5 u^{2} + 8 u + 3 = 0

Quadratic Equation Formula:

For

a = 5, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

8^{2} - 4 \cdot 5 \cdot 3 = 2

8^{2} - 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

Subtract the numbers:

64 - 60 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- 8 \pm 2}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- 8 + 2}{2 \cdot 5}, u_{2} = \frac{- 8 - 2}{2 \cdot 5}

u = \frac{- 8 + 2}{2 \cdot 5} : - \frac{3}{5}

\frac{- 8 + 2}{2 \cdot 5}

Add/Subtract the numbers:

- 8 + 2 = - 6

= \frac{- 6}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 6}{10}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{10}

Cancel the common factor:

2

= - \frac{3}{5}

u = \frac{- 8 - 2}{2 \cdot 5} : - 1

\frac{- 8 - 2}{2 \cdot 5}

Subtract the numbers:

- 8 - 2 = - 10

= \frac{- 10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{10}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = - \frac{3}{5}, u = - 1

Substitute back

u = cot (x)

cot (x) = - \frac{3}{5}, cot (x) = - 1

cot (x) = - \frac{3}{5}, cot (x) = - 1

cot (x) = - \frac{3}{5} : x = arccot (- \frac{3}{5}) + πn

cot (x) = - \frac{3}{5}

Apply trig inverse properties

cot (x) = - \frac{3}{5}

General solutions for

cot (x) = - \frac{3}{5}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{3}{5}) + πn

x = arccot (- \frac{3}{5}) + πn

cot (x) = - 1 : x = \frac{3 π}{4} + πn

cot (x) = - 1

General solutions for

cot (x) = - 1

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = arccot (- \frac{3}{5}) + πn, x = \frac{3 π}{4} + πn

Show solutions in decimal form

x = 2.11121 \dots + πn, x = \frac{3 π}{4} + πn

Popular Examples

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2tan^2(2x)-tan(2x)=3

2 tan^{2} (2 x) - tan (2 x) = 3

sin(x)= 15/31

sin (x) = \frac{15}{31}

4cot(θ)-1=0

4 cot (θ) - 1 = 0

solvefor x,4cos^2(x)=5-4sin(x)solve for

x, 4 cos^{2} (x) = 5 - 4 sin (x)

Frequently Asked Questions (FAQ)

What is the general solution for 5cot^2(x)+8cot(x)+3=0 ?
The general solution for 5cot^2(x)+8cot(x)+3=0 is x=2.11121…+pin,x=(3pi)/4+pin